## Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees

Venue: | Trans. Amer. Math. Soc |

Citations: | 17 - 6 self |

### BibTeX

@ARTICLE{Cholak_automorphismsof,

author = {Peter Cholak and Richard Coles and Rod Downey and Eberhard Herrmann},

title = {Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees},

journal = {Trans. Amer. Math. Soc},

year = {},

volume = {353},

pages = {4899--4924}

}

### OpenURL

### Abstract

Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin’s work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare’s work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of Π0 1 classes) forms an orbit in the lattice of Π01 classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of Π0 1 classes. We remark that the automorphism result is proven via a ∆0 3 automorphism, and demonstrate that this complexity is necessary. 1.

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Citation Context ...if I is a computable ideal in a computable ring R then the collection of prime ideals containing I forms a Π 0 1 class. Π 0 1-classes have been studied for many years (for example, Jockusch and Soare =-=[18]-=-, Kreisel [19]) and are particularly associated with models of Peano arithmetic (PA) and proof theoretical notions such as WKL0. There are many ways of viewing Π0 1 classes and many connections of com... |

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Citation Context ...ove). This would require significant technology, since there are computable Boolean algebras that are not even arithmetically isomorphic. This is even true of rank 1 Boolean algebras (Downey-Jockusch =-=[11]-=-). A good test case is to consider whether, when B1 and B2 are computable copies of the Boolean algebra of finite and cofinite sets, are B1 and B2 automorphic? 3 The proof technique from Theorem 7.6 i... |

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Citation Context ...computable automorphism of Q. Moreover, we can construct two thin Π0 1-classes that are not automorphic via a ∆0 2 automorphism. This seems to present an obstacle to our program. Nevertheless, Remmel =-=[24]-=- has proved that L(2ω )has2ℵ0automorphisms. We give a proof of Remmel’s theorem in section 6. So there is some hope of an analogue of Soare’s theorem. Indeed, as we show, this is the case. Using some ... |

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Citation Context ...is, each follower needs more permissions than its predecessor for followers of the same requirement. Eventually a new degree class, called the anc degrees, was introduced by Downey, Jockusch and Stob =-=[12]-=- to explain such arguments. It turns out that the anc degrees are a class containing all nonlow2 degrees, and are closed upwards. They are exactly the degrees realized by many known constructions. We ... |

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Citation Context ...esults for constructing automorphisms of L(2ω ). Evidence from other structures said that the presence of additional algebraic structure can lead to quite different situations. For instance, Guichard =-=[15]-=- proved that the lattice of c.e. subspaces of an infinite dimensional fully effective vector space L(V∞) has only countably many automorphisms, each induced by a computable semi-linear transformation ... |

1 | A maximal set that is not complete - Sacks - 1964 |